Settlement Calculation of Semi-Rigid Pile Composite Foundation on Ultra-Soft Soil under Embankment Load

Abstract

Ultra-soft soil is distributed in coastal areas around the world and has poor engineering properties. There is a significant difference in settlement between semi-rigid pile and surrounding soil under embankment load. Based on existing research results, the settlement calculation formula of ultra-soft soil composite foundation reinforced by semi-rigid pile is derived in this paper. Based on the Alamgir displacement model, assuming a three-zone model of pile skin friction with a negative skin friction plastic zone in the upper part of the pile, an elastic zone in the middle part of the pile, and a skin friction-bearing plastic zone in the lower part of the pile, the upward and downward penetrations of pile, and pile–soil slip deformation characteristics are considered. Analytical expressions for settlement calculations of semi-rigid pile composite foundations under embankments were derived based on differential equations for pile–soil load transfer in the unit cell. The influences of pile diameter and the compression modulus of the underlying layer at the pile end on the settlement characteristics of the semi-rigid pile composite foundation are discussed. The results show that the derived theoretical calculation method is in good agreement with the field measurement and laboratory model test results. Ultra-soft soil composite foundations have long settlement stabilization times and large settlement deformations. Penetration deformation occurs at the semi-rigid pile end. The relationship between pile end resistance and pile end piercing deformation is hyperbolic. The compression modulus of the underlying layer has a great influence on pile end penetration. The lower the compression modulus of the underlying layer, the larger the penetration deformation of pile end. The larger the pile diameter is, the smaller the penetration deformation is.

1. Introduction

Ultra-soft soil, as a special property soil [1,2,3], usually occurs as sedimentary layers in coastal areas, mainly in coastal areas such as the eastern coast of China [4,5,6], Ariake Bay of Japan [7,8], the eastern coast of Singapore [1,9,10], and Bangkok Plain [11,12]. Most of them were formed in the late Quaternary [13,14,15] and have poor engineering characteristics, such as high water content, high compressibility, low permeability, and low strength [2,3,9,10,16]. When constructing railways and highways on ultra-soft ground, the foundation soil will have large settlement deformation and lateral displacement, and embankment collapse accidents occur very easily, resulting in significant economic losses [8,17,18,19]. In order to ensure the stability of embankment and control the settlement after construction, it is necessary to reinforce the ultra-soft ground foundation. Deep cement-mixed (DCM) piles (known as semi-rigid piles) are an effective soft foundation treatment measure with the advantages of low cost, fast construction progress, and significant settlement control [8,11,20,21,22], and they have been widely used around the world, for example, in highway and railway embankments in China [23,24,25], road and railway embankments in Japan [8,17,26], and highway embankments and levees in the USA [27,28]. In most studies so far, the behavior of semi-rigid piles reinforcing soft ground is mainly studied through field tests, model tests, and numerical simulations [22,29,30,31,32,33], while there are fewer theoretical studies on settlement calculation of semi-rigid pile composite foundations.

It has been confirmed by model tests and field measurement data that large differential settlements occur between the DCM piles and the surrounding soil under embankment loading [27,34,35,36,37,38,39]. Xu et al. [40] found that the pile–soil differential settlement of end-bearing DCM piles occurred from the beginning of filling. Floating DCM piles showed differential pile–soil settlement after the embankment load reached a certain height. The pile–soil differential settlement of the end-bearing DCM piles was 20–60 mm, and that of the floating DCM piles was about 150 mm. Dehghanbanadaki et al. [31] found, through 1 g indoor model tests, that when the stiffness difference between pile and surrounding soil is large, especially in floating DCM piles, the pile end is prone to penetration deformation. The above indicates that the skin friction of pile is exerted to its limit due to large pile–soil relative displacement at the pile top and end zone. Therefore, for ultra-soft soil composite foundation reinforced by semi-rigid piles, the nature of surrounding soil is even worse, the skin friction of pile can be exerted to the limit value very easily, and there is relative plastic slip of pile–soil in the pile top and end zone.

The settlement problem of composite foundation has been the focus for geotechnical engineers, and many scholars have studied the settlement calculation of composite foundation. Alamgir et al. [41] proposed a one-dimensional vertical displacement model that can characterize the inhomogeneous settlement of soil between piles by adopting a unit cell model, and they derived analytical equations for the calculation of pile stress, skin friction of pile, and settlement in the reinforced zone. It provides a new way of thinking for the settlement calculation of composite foundation under flexible foundation. However, its displacement mode is only applicable to cushionless composite foundation with a rigid underlying layer at the pile end. Yang [42] established a one-dimensional vertical displacement model of the surrounding soil that can reflect the phenomenon of negative skin friction in pile based on the displacement model of Alamgir et al. [41]. Liu et al. [43] established a two-dimensional displacement model, considering the vertical and radial deformation of surrounding soil. However, they [42,43] assumed that the displacement of the pile–soil interface was coordinated and did not consider the relative slip between them. It is assumed that the pile top stress is equal to the surrounding soil top stress, and the interaction between pile–soil–cushion layer is neglected. Zhang et al. [44] proposed a one-dimensional vertical displacement model soil between piles, considering the penetration deformation of pile top and end for stiffened deep mixed piles, assuming that the skin friction of pile is linearly distributed along the depth. Lu et al. [45] modified the Alamgir displacement model to consider the existence of relative movement at the pile–soil interface, assuming that the skin friction of pile is linearly distributed along the depth, and derived an analytical solution for the settlement calculation of composite foundation. Chen et al. [46] proposed an analytical solution for settlement calculation in a one-dimensional vertical displacement mode by using a unit cell model assuming a three-section distribution of skin friction, considering embankment soil arch, and penetration deformation of pile top and end. However, it does not consider the nonuniform settlement characteristics of the soil between piles. Liu [47] derived an analytical solution for the settlement of composite foundations under flexible foundations by assuming a linear distribution of skin friction along the depth on the basis of the improved displacement model by Yang [42]. Chen et al. [48] considered the critical pile length of flexible piles, the upward and downward penetrations of pile, and assumed a bilinear distribution of skin friction along the depth. Based on the Alamgir displacement model, a settlement calculation formula for flexible pile composite foundation was derived for easy engineering application according to the pile–soil-cushion layer coordination relationship. However, they [47,48] did not consider the plastic slip between the pile–soil when the skin friction of pile reaches the limit value.

Based on the above study, this paper improves the Alamgir displacement model by assuming a three-zone model of pile skin friction with a negative skin friction plastic zone in the upper part of the pile, an elastic zone in the middle part of the pile, and a skin friction-bearing plastic zone in the lower part of the pile. Consider the upward and downward penetrations of semi-rigid pile and pile–soil slip deformation characteristics. Analytical expressions for settlement calculations of semi-rigid pile composite foundations under embankments are derived from the differential equations for cell load transfer. And the calculation method of this paper is also verified by combining field tests and model tests. The influence of the factors of pile diameter and the compression modulus of the underlying layer at the pile end on the settlement characteristics of the semi-rigid pile composite foundation is explored, and suggestions are made for its application. We provide reference for the semi-rigid pile–ultra-soft soil composite foundation engineering design.

2. Analytical Model

2.1. Basic Assumptions

The working properties of ultra-soft soil composite foundation are more complicated. In this paper, the unit cell model is used for calculation, and the following assumptions are made.

(1)

The embankment filling is simplified as a uniform load p, and after the adjustment of the cushion layer, it acts on the top surface of the pile and the surrounding soil in the form of a uniform load p_p and p_s.

(2)

The pile and the surrounding soil are isotropic linear elastomers. The elastic modulus, Poisson’s ratio, and diameter of the pile are constant. The compression modulus and Poisson’s ratio of the surrounding soil are constant.

(3)

The pile stress distribution is uniform, and the settlement is the same in the same horizontal plane of composite foundation. The stress and settlement of the surrounding soil at the same horizontal plane are nonuniform. Only vertical deformation occurs in the pile and surrounding soil, and lateral deformation is negligible. Only one-dimensional uniform compression occurs in the underlying layer.

(4)

Pile top load and upward penetration deformation conform to a linear relationship. The relationship between pile end resistance and downward penetration deformation is hyperbolic.

2.2. Model of the Unit Cell

The single pile and the soil within its reinforcement area are taken as concentric cylinders with an equal area equivalent to the computational unit cell for this study. The unit cell model can characterize the force and deformation characteristics of the pile group composite foundation, especially near the middle reinforced area of the roadbed; this is a simplified model commonly used in the calculation of composite foundation. As shown in Figure 1a, the radius of a pile is a and the equivalent radius of a single pile reinforcement is b. As an example, a square triangle is used to lay out the piles, which can be calculated using the Equation (1) proposed by Balaam et al. [49].

$$b=0.5c_{g}S,$$

(1)

where S is the pile spacing; c_g is the influence coefficient of shape of laying-out piles, which are 1.05, 1.13, and $1.13\sqrt{S_1{S}_2}$ (S₁ and S₂ are pile longitudinal and transverse spacing, respectively) for square triangular, square, and rectangular pile laying-out, respectively.

2.3. Mechanism of Pile–Soil–Cushion Interaction

The composite foundation system consists of four main components: embankment, cushion layer, pile reinforcement zone, and underlying layer. For ultra-soft soil foundation, soil properties are poor under embankment loading, since the compression modulus of the pile is much larger than that of the soil between the piles, and the compressive deformation of the surrounding soil is larger than that of the pile. As a result, the displacement of the soil between the piles at the top plane of the pile is larger than the displacement of the pile. The stress concentration effect occurs at the top of pile. As the load increases, the stress concentration effect of the pile becomes more obvious. Due to the stress incongruity caused by the difference in pile–soil settlement, the top of the pile begins to penetrate into the cushion layer to coordinate the stress incongruity caused by the difference in pile–soil settlement. Thereby, a negative skin friction zone appears at the top of the pile and a neutral plane with zero friction at a certain depth below the top of the pile. The soil between piles above the neutral plane moves downwards relative to the pile and exerts negative skin friction on the pile, while the surrounding soil below the neutral plane moves upward relative to the pile and the pile is subjected to positive skin friction. With the further increase in load, the skin friction of the lower of pile is further exerted. When the skin friction is fully exerted to the limit state, the pile end starts to produce penetration deformation, and the pile end resistance can be exerted. A load transfer and deformation diagram of pile–soil system is shown in Figure 2.

According to Figure 2, the settlement deformation coordination equation of all parts of the ultra-soft soil composite foundation system can be obtained as follows:

$$w_s=s_{s1}-s_{s2}$$

(2)

$$w_p=s_{p1}-s_{p2}$$

(3)

$$s_{s1}=s_{p1}+{\delta }_{up}$$

(4)

$$s_{s2}=s_{p2}-{\delta }_{down}$$

(5)

where w_s and w_p are the total compressions of surrounding soil and pile, respectively; s_s1 and s_s2 are the settlements of the surrounding soil surface and top surface of the underlying layer, respectively; s_p1 and s_p2 are the settlements of pile top and pile end, respectively; and δ_up and δ_down are the penetrations of pile top and pile end, respectively.

Through Equations (2)–(5), we can obtain the relationship equation between the surrounding soil compression and the compression of pile, the upward and downward penetrations of piles as follows:

$$w_s=w_p+{\delta }_{up}+{\delta }_{down}$$

(6)

The settlement of piles and surrounding soil at the neutral plane are equal, and the pile–soil deformation coordination equations above and below the neutral plane can be obtained from Equation (6), respectively:

$$w_{su}=w_{pu}+{\delta }_{up}$$

(7)

$$w_{sd}=w_{pd}+{\delta }_{down}$$

(8)

where w_su and w_pu are the compressions of the surrounding soil and pile above the neutral plane, respectively; w_sd and w_pd are the compressions of the surrounding soil and pile below the neutral plane, respectively.

3. Analysis of Pile–Soil Interaction

3.1. Skin Friction Distribution Model

The degree of skin friction exerted depends on the magnitude of the pile–soil relative displacement, and it increases with increasing relative pile–soil displacement up to the limiting value. The relative displacement of the pile–soil when the skin friction is exerted to its ultimate value is called the ultimate relative displacement, which is related to the property of the soil. This relationship, which reflects the relationship between skin friction and pile–soil relative displacement, is often referred to as the load transfer function. The load transfer function is classified into an ideal elastic–plastic model, bilinear model [50], hyperbolic model [51], and trilinear model [52]. Among them, the ideal elastic–plastic model reflects the shear properties of pile–soil interface in a more concise way and has been widely used [45,46,48,53]. In this paper, the ideal elastic–plastic model is selected to simulate the relationship between skin friction and pile–soil relative displacement, as shown in Figure 3. The interfacial shear stiffness K_s was calculated using the empirical Equation (9) given by Randolph et al. [54]. The application assumes that the soil shear modulus G_s is constant along the depth.

$$K_s=\frac{{\tau }_u(z)}{{\delta }_u(z)}=\frac{2G_{s}l}{a^2\ln (r_m/a)}$$

(9)

where δ_u is the ultimate relative displacement of pile–soil at depth z; τ_u is the ultimate skin friction at depth z; l is the length of the pile; r_m is the radius of influence of a single pile, which is recommended to be taken as $r_m=2.5\rho l(1-{\nu }_s)$ in the literature [54], where ρ is the ratio of shear modulus between soil in the reinforced area and soil at the pile end.

The ultimate skin friction varies along the depth and is calculated according to Equation (10) [44,55,56,57].

$${\tau }_u(z)=c_a+K_0\gamma z\tan {\phi }_a$$

(10)

where c_a and φ_a are the cohesion and internal friction angles of the pile–soil interface, respectively; γ is the weight of soil; and K₀ is horizontal earth pressure coefficient in the reinforced area, which is calculated by $K_0=1-\sin {\phi }_a$ [58]. Let $A=c_a$ and $B=K_0\gamma \tan {\phi }_a$.

According to the analysis of the pile–soil–cushion interaction mechanism under embankment loading in Section 2.3, and the research results of Chen et al. [46], a three-zone model of pile lateral friction resistance is established in this paper, as in Figure 4. That is, 0~l₁ is the negative skin friction plastic zone (Ⅰ), and the negative skin friction reaches the limit value; l₁~l₀ is the negative skin friction elastic zone (Ⅱ); l₀~l₂ is the positive skin friction elastic zone (Ⅱ); l₂~l is the positive skin friction plastic zone (Ⅲ), and the positive skin friction reaches the limit value; and l₀ is the location of the neutral plane of pile.

An expression for the pile–soil relative displacement can be obtained from Figure 4 as:

$$\delta (z)=\left\{\begin{array}{ll}\frac{l_0-z}{l_0-l_1}{\delta }_{u1}& \left(0\le \mathrm{z}\le l_0\right)\\ \frac{z-l_0}{l_2-l_0}{\delta }_{u2}& \left(l_0\le \mathrm{z}\le l\right)\end{array}\right.$$

(11)

where δ_u1 and δ_u2 are the ultimate pile–soil relative displacement in the negative and positive skin friction elastic zones, respectively.

The distribution function of skin friction is as follows:

$$\tau (z)=\left\{\begin{array}{ll}-A-Bz& \left(0\le z\le l_1\right)\\ \frac{z-l_1}{l_2-l_1}\left({\tau }_2+{\tau }_1\right)-{\tau }_1& \left(l_1\le z\le l_2\right)\\ A+Bz& \left(l_2\le z\le l\right)\end{array}\right.$$

(12)

where τ₁ and τ₂ are the ultimate skin friction at depths l₁ and l₂, respectively.

3.2. Pile End Load Transfer Model

The load transfer model is used to reflect the relationship between the pile end resistance and the pile end penetration deformation. At present, there is no reasonable calculation theory to describe the performance behavior of pile tip resistance with penetration deformation. To solve this problem, scholars have mainly relied on field tests [59], model tests [60,61,62], and finite element or numerical analysis of particle flow code [63,64]. Based on the results of field and model tests, Hirayama et al. [65], Yasufuku et al. [66], Bohn et al. [67], Madhira et al. [68], and Vedhasri et al. [69] have used hyperbolic models to simulate the development of pile end resistance with penetration deformations, which were in better agreement with the test results. Moreover, the hyperbolic model is simpler than other curves in calculating and predicting pile end resistance [70].

However, the hyperbolic model does not reflect the softening characteristics of end resistance and cannot reflect the post-peak effect of ultimate end resistance. Examples of conditions are the occurrence of piercing damage at the pile end and degradation under cyclic loading. Single-pile static load tests have shown that for piles without puncture damage at the pile end, the load–displacement relationship at the pile end exhibited work-hardening characteristics [71]. Based on the previous research results, the hyperbolic model was selected for pile end load transfer in the paper, i.e., the pile end resistance p_b and penetration deformation δ_down satisfy Equation (13).

$$p_b=\frac{{\delta }_{down}}{\frac{1}{k_b}+\frac{1}{p_{bu}}{\delta }_{down}}$$

(13)

where k_b is the initial stiffness of pile end soil, which is taken as $k_b=4G_{sb}/\pi a(1-{\nu }_{sb})$ according to the recommendations of Randolph et al. [54,72]; G_sb and ν_sb are the shear modulus and Poisson’s ratio of the underlying soil, respectively; and p_bu is the ultimate pile end resistance.

From Equation (13), it can be seen that the ultimate pile end resistance is the key parameter for calculation. At present, the research on the ultimate pile end resistance is mainly based on the most extensive application of cavity expansion theory. The results obtained are also different with different constitutive relations. Domestic and foreign scholars have proposed a series of analytical or semi-analytical theoretical solutions for cylindrical/spherical cavity expansion by using elasto-plastic soil constitutive models such as the Mohr–Coulomb model and the modified Cam-Clay model [73,74]. However, as the complexity of the constitutive model increases, the determination of material parameters is more cumbersome. This limits the engineering application of the relevant theoretical methods. In this paper, the Mohr–Coulomb yield criterion with relatively few parameters is used to simulate the pile end piercing into the holding layer as a spherical cavity expansion process. Based on cavity expansion theory, the ultimate pile end resistance equation is derived.

It is assumed that the initial radius of the spherical cavity is R₀ at one point in the soil in infinite space, and expansion occurs under the action of the uniform pressure of the spherical cavity, which is called the expansion pressure p_b. When the pressure is small, the soil around the cavity is in an elastic state. As the pressure continues to increase, the soil around cavity from the elastic state gradually enters into the plastic state. As the value of p_b increases, the plastic zone expands until the pressure p_b increases to the limiting value, i.e., the ultimate pressure p_bu. At this point, the radius of the spherical cavity is R_u. The radius of the elastic–plastic junction is R_p. Within the radius R_p is the plastic zone, beyond which the soil remains elastic. The spherical cavity expansion model is shown in Figure 5.

Since spherical cavity expansion is a centrosymmetric problem, using spherical coordinates, the equilibrium differential equation is Equation (14).

$$\frac{d{\sigma }_r}{dr}+2\frac{{\sigma }_r-{\sigma }_{\theta }}{r}=0$$

(14)

It is assumed that the underlying soil obeys the Mohr–Coulomb yield criterion. Then, the yield condition of the Mohr–Coulomb material is given as Equation (15) [75].

$${\sigma }_r-{\sigma }_{\theta }=\left({\sigma }_r+{\sigma }_{\theta }\right)\sin \phi +2c_b\cos {\phi }_b$$

(15)

where σ_r, σ_θ are radial and circumferential stresses, respectively; r is the distance from the centre of the spherical cavity; E_b, ν_b are the elastic modulus and Poisson’s ratio of soil in the pile end, respectively; c_b, φ_b are the cohesion and the internal friction angle of soil, respectively.

According to the inverse solution of the theory of elasticity [76], it is assumed that the stress function ψ is only a function of the radial coordinate r. The stress function ψ is given as Equation (16).

$$\psi =Z\ln r,$$

(16)

Thus, the radial and circumferential stresses at any point are Equation (17).

$$\left\{\begin{array}{l}{\sigma }_r=\frac{1}{r}\frac{d\psi }{dr}=\frac{Z}{r^2}\\ {\sigma }_{\theta }=\frac{d^2\psi }{d{r}^2}=-\frac{Z}{r^2}\end{array}\right.$$

(17)

The constant Z is obtained by substituting the boundary condition r = R₀ with σ_r = p_b into Equation (17). Combining the geometrical equations and the constitutive equations of the spherical coordinates, the expressions of stress and displacement in the elastic phase are obtained as Equations (18)–(20), respectively.

$${\sigma }_r=\frac{{R_0}^2{p}_b}{r^3},$$

(18)

$${\sigma }_{\theta }=-{\sigma }_r,$$

(19)

$$u_r=\frac{r}{2G_b}{\sigma }_r,$$

(20)

where G_b is the shear modulus of the soil at the pile end, calculated as G_b = E_b/[2(1 + ν_b)].

Substituting the yield condition Equation (15) into the equilibrium differential Equation (14) leads to Equation (21).

$$\frac{d{\sigma }_r}{dr}+\frac{4\sin {\phi }_b}{1+\sin {\phi }_b}\frac{{\sigma }_r}{r}+\frac{4c_b\cos {\phi }_b}{\left(1+\sin {\phi }_b\right)}\frac{1}{r}=0$$

(21)

Equation (21) is a first-order linear differential equation, which is solved and brought to the final expansion radius boundary condition r = R_u when σ_r = p_bu, which leads to the plastic zone stress expressions Equations (22) and (23) [77,78].

$${\sigma }_r=\left(p_{bu}+c_b\cot {\phi }_b\right){\left(\frac{R_u}{r}\right)}^{\frac{4\sin {\phi }_b}{1+\sin {\phi }_b}}-c_b\cot {\phi }_b$$

(22)

$${\sigma }_{\theta }={\sigma }_r\frac{1-\sin {\phi }_b}{1+\sin {\phi }_b}-\frac{2c_b\cos {\phi }_b}{1+\sin {\phi }_b}$$

(23)

According to Equations (15) and (19), the radial stress at the elastic–plastic junction r = R_p can be obtained as Equation (24).

$${\sigma }_p=c_b\cos {\phi }_b,$$

(24)

Substituting Equation (24) into Equation (20) yields the radial displacement u_p at the elastic–plastic junction as Equation (25).

$$u_p=\frac{R_p}{2G_b}{c}_b\cos {\phi }_b,$$

(25)

From the soil stress continuity condition, i.e., the radial stresses are equal at the elastic–plastic interface, the joint Equations (22) and (24) can be obtained as Equation (26).

$$\left(p_{bu}+c_b\cot {\phi }_b\right){\left(\frac{R_u}{R_p}\right)}^{\frac{4\sin {\phi }_b}{1+\sin {\phi }_b}}-c_b\cot {\phi }_b=\frac{4c_b\cos {\phi }_b}{3-\sin {\phi }_b}$$

(26)

The expansion process of the cavity can be regarded as an undrained process and the volume strain of the soil around cavity is zero [46]. Ignoring the volume change in plastic materials in the elastic phase, the volume of soil displaced during the expansion process is equal to the volume change in the plastic zone. Then, Equation (27) can be obtained.

$${R_u}^3-{R_0}^3={R_p}^3-\left({R_p}^3-{u_p}^3\right)$$

(27)

Expanding Equation (27) and omitting the higher terms of u_p and R₀, associating Equation (25) yields Equation (28).

$$\frac{R_u}{R_p}={\left[\frac{3c_b\cos {\phi }_b}{2G_b}\right]}^{\frac{1}{3}},$$

(28)

Substituting Equation (28) into Equation (26) yields the ultimate expansion pressure, i.e., the ultimate pile end resistance, which is calculated by the expression Equation (29).

$$p_{bu}=\left(\frac{4c_b\cos {\phi }_b}{3-\sin {\phi }_b}+c_b\cot {\phi }_b\right){\left[\frac{2G_b}{3c_b\cos {\phi }_b}\right]}^{\frac{4\sin {\phi }_b}{3\left(1+\sin {\phi }_b\right)}}-c_b\cot {\phi }_b$$

(29)

4. Settlement Calculation of Composite Foundation

4.1. Compressive Deformation of Reinforcement Area

To carry out deformation analysis of the reinforced area, the vertical additional stress distribution of the surrounding soil needs to be obtained. The surrounding soil of composite foundation under flexible foundation, such as embankment, undergoes nonuniform compressive deformation, and the vertical additional stresses are different in the same plane.

4.1.1. Vertical Displacement Model of Surrounding Soil

In order to reflect the different settlement characteristics of soil between piles in the same horizontal plane of composite foundation under embankment load, the vertical displacement model of surrounding soil was improved based on Alamgir et al. [41]. As shown in Equation (30):

$$w_s(r,z)=w_p(z)+g(z)\left[\frac{r}{a}-e^{\beta \left(\frac{r}{a}-1\right)}\right]+f(z)$$

(30)

where w_s(r,z) is the vertical displacement of the surrounding soil at any point; w_p(z) is the vertical displacement of the pile at depth z; a is the radius of the pile; r is the horizontal distance from the centre of the pile; β is a coefficient of determination to be made; and g(z) and f(z) are functions of depth z.

Ignoring the radial displacement of the surrounding soil, the shear strain of soil at any point (r,z) between piles can be obtained from the geometric equation of elastic theory as Equation (31).

$${\gamma }_s=\frac{𝜕w_s(r,z)}{𝜕r},$$

(31)

The shear stress at any point of surrounding soil is as follows:

$${\tau }_s(r,z)=G_s{\gamma }_s,$$

(32)

where G_s is the shear modulus of the surrounding soil.

Associative Equations (30)–(32) lead to Equation (33).

$${\tau }_s(r,z)=\frac{G_{s}g(z)}{a}\left[1-\beta e^{\beta \left(\frac{r}{a}-1\right)}\right]$$

(33)

At the boundary of the calculation unit body, the skin friction is zero, i.e., ${\tau }_s(r,z){\mid }_{r=b}=0$ when r = b. Substituting this into Equation (33) yields Equation (34).

$$1-\beta e^{\beta \left(\frac{b}{a}-1\right)}=0$$

(34)

From Equation (34), the value of constant β can be obtained from the pile design parameters of the composite foundation.

When r = b, the skin friction can be calculated by Equation (35).

$$\tau (z)={\tau }_s(r,z){\mid }_{r=a}=\frac{G_s\left(1-\beta \right)}{a}g(z)$$

(35)

From Equation (35), g(z) differs from τ(z) by a constant, so g(z) can be understood as the distribution function of skin friction.

Associative Equations (12) and (35) give g(z) as Equation (36).

$$g(z)=\left\{\begin{array}{ll}-\frac{a\left(A+Bz\right)}{G_s\left(1-\beta \right)}& \left(0\le \mathrm{z}\le l_1\right)\\ \frac{a}{G_s\left(1-\beta \right)}\left[\frac{z-l_1}{l_2-l_1}\left({\tau }_1+{\tau }_2\right)-{\tau }_1\right]& \left(l_1\le \mathrm{z}\le l_2\right)\\ \frac{a\left(A+Bz\right)}{G_s\left(1-\beta \right)}& \left(l_2\le \mathrm{z}\le l\right)\end{array}\right.$$

(36)

Substituting r = a into Equation (30), the vertical displacement function of surrounding soil becomes the pile–soil interface slip function.

$$w_s(r,z)-w_p(r,z)=f(z)=\delta (z)$$

(37)

From Equation (37), f(z) can be interpreted as the slip function at the pile–soil interface along the depth z of the foundation.

4.1.2. Establishment and Solution of Differential Equations for the Unit Cell

The composite foundation unit is stressed, as shown in Figure 1b. The micro-section unit with cylindrical pile thickness dz and surrounding soil thickness dz is taken as the object of study, and the forces are shown in Figure 1c.

The equilibrium differential equation is given by the vertical force equilibrium condition of the pile unit as Equation (38).

$$\pi a^{2}d{\sigma }_p(z)-\pi a^2\left[{\sigma }_p(z)+d{\sigma }_p(z)\right]-2\pi a\tau (z)dz=0$$

(38)

simplification gives:

$$\frac{d{\sigma }_p(z)}{dz}=-\frac{2}{a}\tau (z),$$

(39)

The equilibrium differential equation is given by the vertical force equilibrium condition of the soil unit between piles as Equation (40).

$${\sigma }_s(r,z)\cdot \left[\pi {(r+dr)}^2-\pi r^2\right]+2\pi r{\tau }_s(r,z)dz-2\pi (r+dr)\left[{\tau }_s(r,z)+d{\tau }_s(r,z)\right]dz-\left[{\sigma }_s(r,z)+d{\sigma }_s(r,z)\right]\left[\pi {(r+dr)}^2-\pi r^2\right]=0$$

(40)

By omitting the higher order minutiae, Equation (40) can be rewritten as follows:

$$\frac{𝜕{\sigma }_s(r,z)}{𝜕z}=-\left(\frac{{\tau }_s(r,z)}{r}+\frac{𝜕{\tau }_s(r,z)}{𝜕r}\right)$$

(41)

Substituting Equation (33) into Equation (41) yields Equation (42).

$$\frac{𝜕{\sigma }_s(r,z)}{𝜕z}=-\frac{G_{s}g(z)}{a}\left[\frac{1}{r}-\left(\frac{1}{r}+\frac{\beta }{a}\right)\beta e^{\beta \left(\frac{r}{a}-1\right)}\right]$$

(42)

Let $H(r)=\frac{1}{1-\beta }\left[\frac{1}{r}-\left(\frac{1}{r}+\frac{\beta }{a}\right)\beta e^{\beta \left(\frac{r}{a}-1\right)}\right]$, then Equation (42) is simplified as follows:

$$\frac{𝜕{\sigma }_s(r,z)}{𝜕z}=-\frac{G_{s}g(z)H(r)}{a}\left(1-\beta \right)$$

(43)

It can be seen that the change in the vertical stress of the surrounding soil along the depth is closely related to the distribution function of skin friction. The joint Equations (35) and (43) can be obtained as Equation (44).

$$\frac{𝜕{\sigma }_s(r,z)}{𝜕z}=-H(r)\tau (z),$$

(44)

Substituting Equation (12) into Equations (39) and (44) and solving the first-order differential equations, the vertical stresses of pile and surrounding soil can be obtained as Equations (45) and (46), respectively.

$${\sigma }_p(z)=\left\{\begin{array}{ll}\frac{2}{a}Az+\frac{1}{a}B{z}^2+c_1& \left(0\le z\le l_1\right)\\ \frac{2\left({\tau }_1{l}_2+{\tau }_2{l}_1\right)}{a\left(l_2-l_1\right)}z-\frac{{\tau }_1+{\tau }_2}{a\left(l_2-l_1\right)}{z}^2+c_2& \left(l_1\le z\le l_2\right)\\ -\frac{2}{a}Az-\frac{1}{a}B{z}^2+c_3& \left(l_2\le z\le l\right)\end{array}\right.$$

(45)

$${\sigma }_s(r,z)=\left\{\begin{array}{ll}\frac{B}{2}H(r)z^2+AH(r)z+c_4& \left(0\le z\le l_1\right)\\ \frac{{\tau }_1{l}_2+{\tau }_2{l}_1}{l_2-l_1}H(r)z-\frac{{\tau }_1+{\tau }_2}{2\left(l_2-l_1\right)}H(r)z^2+c_5& \left(l_1\le z\le l_2\right)\\ -\frac{B}{2}H(r)z^2-AH(r)z+c_6& \left(l_2\le z\le l\right)\end{array}\right.$$

(46)

where c₁–c₆ are integration constants.

From the boundary conditions at the top of the pile and surrounding soil, i.e., ${\sigma }_p(z){\mid }_{z=0}=p_p$ and ${\sigma }_s(r,z){\mid }_{z=0}=p_s$ when z = 0, and substituting them into Equations (45) and (46), we obtain c₁ and c₄.

$$c_1=p_p,$$

(47)

$$c_4=p_s,$$

(48)

From the continuity condition of the axial stress of pile and the vertical stress of surrounding soil, i.e., when z = l₁, the segmental functions (0 ≤ z ≤ l₁ and l₁ ≤ z ≤ l₂) of the vertical stress of pile and surrounding soil are equal at the depth of l₁, which leads to c₂ and c₅.

$$c_2=p_p+\frac{2}{a}A{l}_1+\frac{1}{a}B{l_1}^2-\frac{{\tau }_2{l_1}^2+2{\tau }_1{l}_1{l}_2-{\tau }_1{l_1}^2}{a\left(l_2-l_1\right)}$$

(49)

$$c_5=p_s+\frac{1}{2}B{l_1}^{2}H(r)+A{l}_{1}H(r)-\frac{{\tau }_2{l_1}^2+2{\tau }_1{l}_1{l}_2-{\tau }_1{l_1}^2}{2\left(l_2-l_1\right)}H(r)$$

(50)

Similarly, when z = l₂, the pile and surrounding soil vertical stress segmentation function (l₁ ≤ z ≤ l₂ and l₂ ≤ z ≤ l) is equal at depth l₂, which can be obtained as c₃, c₆.

$$c_3=p_p+\frac{2}{a}A\left(l_1+l_2\right)+\frac{1}{a}B\left({l_1}^2+{l_2}^2\right)+\frac{1}{a}\left({\tau }_2-{\tau }_1\right)\left(l_2-l_1\right)$$

(51)

$$c_6=p_s+\frac{1}{2}B\left({l_1}^2+{l_2}^2\right)H(r)+A\left(l_1+l_2\right)H(r)-\frac{1}{2}\left({\tau }_2-{\tau }_1\right)\left(l_2-l_1\right)H(r)$$

(52)

From Equations (45) and (46), the axial strain of the pile and the vertical strain of the surrounding soil can be obtained as Equations (53) and (54), respectively.

$${\epsilon }_p(z)=\left\{\begin{array}{ll}\frac{2}{a{E}_p}Az+\frac{1}{a{E}_p}B{z}^2+C_1& \left(0\le z\le l_1\right)\\ \frac{2\left({\tau }_1{l}_2+{\tau }_2{l}_1\right)}{a{E}_p\left(l_2-l_1\right)}z-\frac{{\tau }_1+{\tau }_2}{a{E}_p\left(l_2-l_1\right)}{z}^2+C_2& \left(l_1\le z\le l_2\right)\\ -\frac{2}{a{E}_p}Az-\frac{1}{a{E}_p}B{z}^2+C_3& \left(l_2\le z\le l\right)\end{array}\right.$$

(53)

$${\epsilon }_s(r,z)=\left\{\begin{array}{ll}\frac{B}{2E_S}H(r)z^2+\frac{A}{E_S}H(r)z+C_4& \left(0\le z\le l_1\right)\\ \frac{{\tau }_1{l}_2+{\tau }_2{l}_1}{E_S\left(l_2-l_1\right)}H(r)z-\frac{{\tau }_1+{\tau }_2}{2E_S\left(l_2-l_1\right)}H(r)z^2+C_5& \left(l_1\le z\le l_2\right)\\ -\frac{B}{2E_S}H(r)z^2-\frac{A}{E_S}H(r)z+C_6& \left(l_2\le z\le l\right)\end{array}\right.$$

(54)

where E_p is the compression modulus of pile body; E_s is the compression modulus of soil between piles; $C_1=c_1/E_p$; $C_2=c_2/E_p$; $C_3=c_3/E_p$; $C_4=c_4/E_s$; $C_5=c_5/E_s$; $C_6=c_6/E_s$.

When the surrounding soil is divided into several layers, E_s is taken as the weighted compression modulus of each layer.

$$E_s=\frac{E_{S1}{h}_1+E_{S2}{h}_2+\cdot \cdot \cdot +E_{Sn}{h}_n}{H}$$

(55)

where E_s1, E_s2, …, E_sn is the compression modulus of each soil layer; h₁, h₂, …, h_n is the thickness of each soil layer in the reinforced area; and H is the total thickness of soil layer in the reinforced area.

From Equations (53) and (54), the compressive deformation of the pile and the surrounding soil above the neutral plane can be obtained as Equations (56) and (57), respectively.

$$w_{pu}={\displaystyle {\int }_0^{l_1}{\epsilon }_p(z)}dz+{\displaystyle {\int }_{l_1}^{l_0}{\epsilon }_p(z)}dz$$

(56)

$$w_{su}={\displaystyle {\int }_0^{l_1}{\epsilon }_s(z)}dz+{\displaystyle {\int }_{l_1}^{l_0}{\epsilon }_s(z)}dz$$

(57)

Similarly, the compressive deformation of the pile and the surrounding soil below the neutral plane can be obtained as Equations (58) and (59) respectively.

$$w_{pd}={\displaystyle {\int }_{l_0}^{l_2}{\epsilon }_p(z)}dz+{\displaystyle {\int }_{l_2}^l{\epsilon }_p(z)}dz$$

(58)

$$w_{sd}={\displaystyle {\int }_{l_0}^{l_2}{\epsilon }_s(z)}dz+{\displaystyle {\int }_{l_2}^l{\epsilon }_s(z)}dz$$

(59)

From Equations (57) and (59), the compressive deformation of composite foundation reinforced area w_s can be obtained.

4.2. The Upward and Downward Penetrations of Pile

Assuming that the soil at the top of the pile conforms to the Winkler foundation model, the pile top penetration is calculated as Equation (60).

$${\delta }_{up}=C_s\left(p_p-p_s\right)$$

(60)

where C_s is the vertical penetration (m/kPa) of cushion under the unit pressure of pile top, i.e., the flexibility coefficient of cushion. It can be calculated according to formula [48] C_s = h_c/E_c, where h_c, E_c are thickness and compression modulus of the cushion layer respectively.

From Equation (13), we can obtain the pile end penetration calculation Equation (61).

$${\delta }_{down}=\frac{p_b}{k_b(1-\frac{1}{p_{bu}}{p}_b)},$$

(61)

where p_b is the pile end resistance, which can be calculated from Equation (45).

4.3. Settlement of the Underlying Layer of Composite Foundation

The key to the calculation of underlying soil settlement is to determine the additional stress at the top of the underlying layer. Substituting z = l into Equation (46), the average additional stress of the surrounding soil at the pile end plane can be obtained as Equation (62).

$$\overline{{\sigma }_s}(r,z){\mid }_{z=l}=\frac{{\displaystyle {\int }_a^b{\sigma }_s(r,z){\mid }_{z=l}\cdot 2\pi rdr}}{\pi \left(b^2-a^2\right)},$$

(62)

The average additional stresses at the top of the underlying layer due to the axial force of the pile and the vertical stresses of the surrounding soil can be obtained by combining Equations (45) and (62).

$${\sigma }_{rs}(z){\mid }_{z=l}=m{\sigma }_p(z){\mid }_{z=l}+\left(1-m\right)\overline{{\sigma }_s}(r,z){\mid }_{z=l}$$

(63)

where m is the area replacement ratio, $m=A_p/\left(A_S+A_p\right)=a^2{/\mathrm{b}}^2$; A_p is the pile top area; and A_s is the area of surrounding soil of the unit cell.

Then, the compression deformation of the underlying layer can be obtained as formula (64).

$$w_{rs}={\psi }_s{\displaystyle \sum _{i=1}^n\frac{{\sigma }_{rs}(z){\mid }_{z=l}}{E_{bi}}}\left({\alpha }_i{z}_i-{\alpha }_{i-1}{z}_{i-1}\right)$$

(64)

where ψ_s is the empirical coefficient of settlement; n is the number of soil layers divided within the calculation depth of the soil deformation in the underlying layer; E_bi is the compression modulus of the ith layer of soil; z_i−1 and z_i are the depths of the top and bottom of the ith layer of soil, respectively; and α_i−1 and α_i are the average additional stress coefficients of the corresponding soil depths.

5. Determination of Calculation Parameters

For application of the established settlement calculation method for the composite foundation of ultra-soft soil under embankment, in addition to the basic conventional parameters, it is necessary to determine the following parameters: composite foundation pile–soil sharing load p_p and p_s; the depth of elastic and plastic zone l₁, l₀, l₂; and ultimate skin friction τ₁, τ₂.

5.1. Pile–Soil Load Sharing

The embankment load is shared by the piles and the surrounding soil. Based on the pile area replacement ratio m, the average surface load of the unit can be obtained as Equation (65).

$$p=m{p}_p+\left(1-m\right)p_s$$

(65)

The ideal elastic–plastic model between skin friction and pile–soil relative displacement can be shown. When z = l₁, the pile–soil relative displacement can be obtained from Equation (9) as follows in Equation (66).

$${\delta }_{u1}=\frac{{\tau }_1}{K_S},$$

(66)

From the distribution mode of pile–soil relative displacement shown in Figure 4a, the following geometric relationship can be obtained.

$$\frac{{\delta }_{u1}}{{\delta }_{up}}=\frac{l_0-l_1}{l_0}$$

(67)

Associative Equations (10), (60), and (65)–(67) lead to Equations (68) and (69).

$$p_s=p-\frac{m{l}_0\left(c_a+K_0\gamma l_1\tan {\phi }_a\right)}{C_s{K}_s\left(l_0-l_1\right)}$$

(68)

$$p_p=p+(1-m)\frac{l_0(c_a+K_0\gamma l_1\tan {\phi }_a)}{C_s{K}_s\left(l_0-l_1\right)}$$

(69)

It can be seen that Equation (68) contains only the unknowns l₁ and l₀ to be found. From Equations (68) and (69), the pile–soil stress ratio n is as follows:

$$n=1+\frac{1}{\frac{C_s{K}_{s}p}{l_0(c_a+K_0\gamma l_1\tan {\phi }_a)}-m}$$

(70)

5.2. Elastic and Plastic Zone Play Depth l₁, l₀, l₂ and τ₁, τ₂

According to the distribution mode of skin friction, as shown in Figure 4b, the following geometric relationship can be obtained.

$$\frac{{\tau }_1}{{\tau }_2}=\frac{l_0-l_1}{l_2-l_0}$$

(71)

From the deformation coordination conditions above and below the neutral plane, Equations (72) and (73) can be obtained.

$$w_{su}-w_{pu}={\delta }_{up}$$

(72)

$$w_{sd}-w_{pd}={\delta }_{down}$$

(73)

Substituting Equations (56)–(59) into Equations (72) and (73), two equations containing only the unknowns l₁, l₀, and l₂ can be obtained. Associating Equations (10) and (71) again, three equations expressed by three unknowns l₁, l₀, l₂ can be obtained. Since these equations are highly nonlinear, simple expressions of the unknowns cannot be obtained directly. These three equations can be solved using the Newton–Raphson method and all other unknowns and variables can be obtained accordingly.

6. Example Validation

6.1. Field Test Validation

The field test site was located at section K14 + 950 of the ultra-soft soil foundation treatment project of Lianyan Railway, Lianyungang City, Jiangsu Province. The field subgrade settlement observation period was 830 days from September 2016 to December 2018. The site overview is as follows: embankment filling thickness 4 m, gravity γ = 20 kN/m³, cushion thickness h_c = 0.3 m, compression modulus E_c = 150 MPa (stress 100–200 kPa), flexibility coefficient c₀ = 2 × 10⁻⁶ m/kPa. The foundation was treated with CDM pile, with pile length l = 16 m, pile radius a = 0.25 m, and pile elastic modulus E_p = 180 MPa, equilar triangle pile layout, pile spacing 1.6 m, single pile treatment equivalent circle radius b = 0.84 m, and replacement rate m = 8.858%. According to the engineering investigation data, the clay layer is distributed in the upper part of the soil layer between piles, which is a crust layer with a thickness of 1.5 m, gravity γ = 17 kN/m³, cohesion c = 12.3 kPa, internal friction angle φ = 4.2°, compression modulus E_s = 3.4 MPa (stress 100–200 kPa), and Poisson’s ratio ν = 0.3. The middle part is composed of Marine ultra-soft soil, which is the main reinforced soil layer with a thickness of 13.5 m, gravity γ = 15.3 kN/m³, cohesion c = 6 kPa, internal friction angle φ = 3.5°, compression modulus E_s = 1.3 MPa (stress 100–200 kPa), and Poisson’s ratio ν = 0.4. The underlying layer is a silt clay layer, gravity γ = 18.5 kN/m³, cohesion c = 28.1 kPa, internal friction angle φ = 12.3°, compression modulus E_s = 7.2 MPa (stress 100~200 kPa), and Poisson’s ratio ν = 0.3. The parameters of each soil layer are shown in

Table 1. Physical and mechanical parameters of field-tested soil.

	Embankment	Cushion Layer	Crust Layer	Ultra-Soft Soil	Underlying Layer
Thickness, h (m)	4	0.3	1.5	13.5	10
Unit weight, γ (kN/m3)	20	21	17	15.3	18.5
Compression modulus, Es1–2 (MPa)	-	150	3.4	1.3	7.2
Poisson’s ratio, ν	-	-	0.3	0.4	0.3
Cohesion, c (kPa)	-	-	12.3	6	28.1
Friction angle, φ (degree)	-	-	4.2	3.5	12.3

.

When the method in this paper is used to calculate the field engineering, the deformation at the center of the soil sampling ring (i.e., r = (a + b)/2 = 0.545 m) is taken as the compressive deformation in the reinforcement area. The calculated parameter β is in one-to-one correspondence with b/a, and β = 0.391 using Equation (34). In addition, the surrounding soil parameters were used as a weighted average of the soil layers within the depth range of the pile length. The cohesion c_a = 1.1 kPa and internal friction angle φ_a = 1.1° at the pile–soil interface were obtained from the shear test at the interface of ultra-soft soil and soil–cement.

The measured central settlement of foundation (CSF) and pile–soil stress ratio (PSSR) for the field test sections and their theoretically calculated values are given in

Table 2. CSF and PSSR.

Projects	CSF (mm)	Relative Error of CSF	Reinforced Area		Underlying Layer		PSSR	Relative Error of PSSR
			Settlement (mm)	Percentage of CSF	Settlement (mm)	Percentage of CSF
Measured value	533.97	-	-	-	-	-	1.75	-
Calculated value	563.00	5.44%	453.60	80.57%	109.40	19.43%	1.25	28.57%

. It can be seen that the CSF measured on site is 533.97 mm, and the settlement of surrounding soil obtained by the calculation method in this paper is 563 mm. Among them, the compression of the reinforcement area is 453.6 mm, accounting for 80.57% of the total settlement, and the compression of the underlying layer is 109.4 mm, accounting for 19.43% of the total settlement. The in-situ PSSR is 1.75, the calculated value is 1.25, and the relative error is 28.57%. It can be seen that the calculated value of the CSF is slightly larger than the measured value, and the relative error is 5.44%. The reason is that the calculated value is the final stability value of soil mass, and the measured value is the final value of the test process. It can be further increased with time to reach the final stability value, which is close to the theoretical calculation result. It shows that the settlement stability time of ultra-soft soil is longer.

The CSF and PSSR obtained by using the theoretical calculation method in this paper are closer to the measured values. It shows that the theoretical calculation method established in this paper is practical for the settlement calculation of ultra-soft soil composite foundation.

6.2. Model Test Validation

6.2.1. Introduction to Model Tests

The model scale of this laboratory model test is 1:50, that is, the ratio of model to prototype. The dimensions of the model box are 1000 (length) × 300 (width) × 500 mm (height), and the box frame is assembled and welded by angle steel. Transparent high-strength tempered glass is used as the boundary constraint around and on the bottom of the model box, which can reduce the friction between the soil and the side wall of the box, and it is conducive to controlling the soil filling.

The model test soil layer is divided into clay, mucky (ultra-soft soil), and silty clay. The bottom of the model box is a silty clay layer with a thickness of 120 mm as the load-bearing layer, two dry densities of 1.2 g/cm³ and 1.4 g/cm³ are considered in the test, and the water content is controlled to be about 16.1%. The mucky layer is 260 mm thick with 83.7% water content, serving as the main reinforcing soil layer. The upper part is a clay layer with a thickness of 60 mm and a water content of 55.3%, which serves as a crust layer. The physical and mechanical parameters of each soil layer are detailed in

Table 3. Physical and mechanical indexes of soil.

Soils	γ (kN/m3)	w (%)	wL (%)	wp (%)	e	Es (MPa)	φ (°)	cq (kPa)
Clay	17.2	55.3	62.4	27.3	1.558	0.581	4.2	12.2
Mucky	15.3	83.7	62.4	27.3	2.173	0.516	3.5	6.0
Silty clay	18.0	16.1	29.6	16.1	0.735	0.8/1 *	12.3	28.1

* The compression modulus of the soil in the table is taken as the value of normal stress from 0 to 50 kPa; 0.8 MPa and 1 MPa are taken for the dry density of silty clay at 1.2 g/cm³ and 1.4 g/cm³, respectively.

.

The model piles were cast with gypsum material. The diameter of the pile was 10 mm and the length of the pile was 320 mm. Gypsum material has low strength, small apparent density, and brittle characteristics. And the strength is easy to control by different ratio combinations, which can reflect the properties of soil–cement materials. It is widely used in model tests [79].

In order to ensure that the model piles can reflect the characteristics of DCM pile, the elastic modulus of the model piles is controlled to be the same as the actual pile modulus. Six different water–gypsum ratios of 0.6, 0.7, 0.8, 1.0, 1.2, and 2.0 were selected to make a Φ50 × 100 mm test sample, and three control samples were prepared for each ratio. After 14 days of maintenance, an unconfined compressive strength test was carried out. The compressive strength and stress–strain curves measured by the compressive strength test are shown in Figure 6. The elastic modulus was calculated from the stress–strain curve, and the elastic modulus corresponding to 50% compressive strength was taken. The change curve of elastic modulus with the water–gypsum ratio is shown in Figure 7. The strength parameters of the samples with different water–gypsum ratios are shown in

Table 4. Mechanical parameters for different water–gypsum ratios.

Water–Gypsum Ratio	Pult (kN)	fult (MPa)	f50 (MPa)	E50 (MPa)
0.6	9.3	4.7	2.4	607
0.7	6.1	3.1	1.6	509
0.8	5.7	2.9	1.5	347
1.0	4.1	2.1	1.0	180
1.2	1.7	0.8	0.4	115
2.0	0.8	0.4	0.2	90

. The water–gypsum ratio w/c = 1.0 was selected as the material ratio of the model pile. The 50% unconfined compressive strength was 1.0 MPa and the elastic modulus was 180 MPa.

The model piles were made by using a hole-in-soil casting method. After the piles were cast, the excavation of the piles was started after 2 h of maintenance. Model piles with initial strengths were obtained, placed horizontally, and cured for 14 days to enhance the pile strength. The preparation process is shown in Figure 8.

Before filling the soil layer, we applied oil on the inside of the model box to reduce the boundary effect. After filling the lower layer, the model pile was first installed, and the remaining soil layer was filled in by layers, ensuring that the pile did not skew during the filling process. After all the soil layers were filled, it was left to stand for 12 h. Micro earth pressure sensors with a measuring range of 0.7 MPa and 0.2 MPa were arranged at the top and end of the pile, respectively. The sensor used an XL2118A static resistance strain tester (Qinhuangdao Xieli Technology Development Co., LTD., Qinhuangdao, China) to collect data. The displacement measurement points were located near the centre of the loading plate and at the pile end. The pile end displacements were measured using 0.2 mm steel wires fixed at the pile ends. The steel wire was connected to the ground surface through a PVC pipe, and the end of the wire was connected to a dial indicator. The accuracy of the indicator was 0.01 mm and the range was 0–30 mm. Before the test loading, a 30 mm sand cushion layer was laid on the top of the soil layer, and then a circular load plate with a diameter of 60 mm was placed. The loading test diagram of semi-rigid pile composite foundation is shown in Figure 9.

6.2.2. Analysis of Model Test Results

Figure 10 gives the variation curves of the soil settlement and the pile end penetration with loading under different conditions of dry density of the underlying layer. From Figure 10, it can be seen that the change curves of the composite foundation loading test load and load plate settlement show a steeply decreasing trend. At the beginning of loading, the difference in settlement deformation under different dry densities is small. With the increase in loading, the difference gradually increases. When the load was increased to 42.44 kPa, the settlement of the underlay with dry density of 1.20 g/cm³ reached 17.897 mm, while the settlement of the dry density of 1.40 g/cm³ was only 6.45 mm (46.69 kPa). At the end of loading, the settlement of the underlying layer with a dry density of 1.20 g/cm³ and 1.40 g/cm³ reached 32.35 mm (49.51 kPa) and 29.26 mm (75.4 kPa), respectively. Substituting the model test parameters into the theoretical calculation method of this paper to obtain the final settlements of both gave 37.04 mm and 34.62 mm, and the relative error of settlement was 18.32% and 15%, respectively. In addition, for the development of pile end penetration at the beginning of loading, the pile end penetration under two dry densities was not obvious. With the increase in loading, the penetration gradually increased, and the final deformations were 19 mm and 13.54 mm, respectively. The pile end penetrations obtained by the theoretical calculation method were 13.03 mm and 11.02 mm, respectively.

The difference in the dry density of the underlying layer reflects, to some extent, the strength and compression modulus of the underlying soil. Through the above analysis, it was found that the smaller the dry density of the underlying layer of the ultra-soft soil composite foundation is, the larger the pile end penetration is, and the more significant the compression deformation of the surrounding soil is. According to the model test results, the ratios of pile end penetration to load plate settlement are 58.73% (1.2 g/cm³) and 46.28% (1.4 g/cm³), and the theoretical calculation values are 35.18% (1.2 g/cm³) and 29.79% (1.4 g/cm³), respectively. It can be seen that the pile end penetration accounts for a relatively large proportion and cannot be ignored. The test value is larger than the theoretical value, and the main reason for this is that the total settlement measured by the test is not the final settlement, and the settlement is relatively small.

Figure 11 shows the development of pile end penetration with pile end resistance. From the Figure 11, it can be seen that in the early loading stage, when the pile end resistance is less than 50 kPa, the penetration is not obvious for the two dry density conditions. With the increase in loading, the pile end penetration of dry density 1.20 g/cm³ develops rapidly. At the final load, the pile end resistances at the two dry densities reached 103.42 kPa and 133.24 kPa, respectively, corresponding to a penetration of 19 mm and 13.54 mm, respectively. The results show that the initial resistance of the pile end is 50 kPa, that is, when the pile end resistance is greater than 50 kPa, pile end penetration will occur. The lower the dry density of the underlying layer, the lower the soil strength of the pile end, the faster the development of the pile end penetration, and the faster the pile end resistance to the extreme value.

Through the theoretical calculation method in this paper, the pile tip resistance and penetration under different loads were obtained. It can be seen that the experimental values are close to the calculated ones. This indicates that the change relationship between pile tip resistance and pile tip penetration is hyperbolic, which is consistent with the research results of Madhira et al. [68] and Vedhasri et al. [69].

6.3. Analysis of Influencing Factors of Pile End Penetration

Through the model test results, it was found that the strength of the underlying layer has a great influence on the pile end penetration. Therefore, combined with the field test parameters, this paper analyzes the influence of the underlying layer compression modulus and pile diameter on the pile end penetration.

Figure 12 shows the curves of pile end penetration with a compression modulus of 4 MPa, 5 MPa, 6 MPa, 7 MPa, 8 MPa, 10 MPa, 12 MPa, 15 MPa, and 20 MPa. From Figure 12, it can be seen that the pile end penetration decreases gradually with the increase in the compression modulus of the underlying layer, and it changes with a power function law. The pile end penetration decreased from 140.2 mm (4 MPa) to 8.23 mm (20 MPa). When the compression modulus of the underlying layer is greater than 10 MPa, the penetration decays slowly. This indicates that the lower the compression modulus of the underlying layer at the pile end, the larger the pile end penetration. In addition, it can be seen that the smaller the pile diameter, the more significant the pile end penetration is when the compression modulus of the underlying layer is low. This indicates that when the compression modulus of the underlying layer is small, the soil strength is low and the pile–soil stiffness is relatively high. Under the same load, the pile end is prone to penetration deformation. On the contrary, when the modulus of the underlying layer increases, the pile–soil stiffness is relatively small, the soil strength is relatively enhanced, the pile end load is transferred to the lower part through the hard soil layer, and the pile end is not easy to penetrate downward. In short, piles with poor end-condition can easily be penetrated downward, while piles with good end-condition are supported in non-soft soil layer and are generally difficult to penetrate downward [40]. The smaller the pile diameter, the relatively larger the load shared by the pile, exacerbating the load on the soil at the pile end. This, coupled with the lower strength of the underlying layer, leads to a significant increase in pile end penetration. It can be inferred that the pile end penetration is large when the composite foundation is designed with floating pile, which is in agreement with the results in the literature [31]. In the design work of composite foundation, the pile end should be placed in the high compression modulus soil layer to reduce the pile end penetration, so as to reduce the compression of the surrounding soil.

Figure 13 shows the relationship curves of pile end penetration at pile diameters of 0.5 m, 0.6 m, 0.7 m, 0.8 m, 0.9 m, and 1 m. From Figure 13, it can be seen that the larger the pile diameter, the smaller the pile end penetration is, and the pile end penetration decreases from 37.61 mm (0.5 m) to 33.89 mm (1 m). When the pile diameter is larger than 0.7 m, the pile end penetration tends to be stable. The calculations in the paper keep the pile spacing constant, and the net distance between piles gradually decreases as the pile diameter increases. When the pile spacing is greater than a certain value, the change in pile end penetration is not obvious, and the main reason for this is related to the range of soil arch effect produced by the pile end. The soil arch effect develops with changes in differential settlements of pile–soil. With the increase in depth, the settlement difference becomes smaller and tends to zero, and the surface with zero settlement difference is equal to the settlement plane. The height of the equal settlement plane is related to the net distance between piles. The smaller the net distance between piles, the smaller the height of the equal settlement plane [80]. In other words, the larger the pile diameter, the smaller the net distance between piles; additionally, the closer the height of the equal settlement plane to the pile end, the smaller the range of soil arch effect at the pile end, the smaller the differential settlement of pile–soil, and the smaller the pile end penetration. On the contrary, the smaller the pile diameter, the larger the net distance between piles; the farther the height of the equal settlement plane from the pile end, the larger the pile end penetration. This shows that the design of composite foundation can reduce the pile end penetration by increasing the pile diameter. And the recommended pile diameter is 0.7 m.

Through the relationship between the compression modulus of the underlying layer, pile diameter, and pile end penetration, it is found that the compression modulus of the underlying layer has a great influence on pile end penetration. In the design of composite foundation, the selection of the lying layer under the pile end should be paid attention to.

7. Conclusions

(1)

For ultra-soft soil composite foundation reinforced by semi-rigid piles, by assuming a linear distribution model in the elastic zone and nonuniform one in plastic zone of skin friction, the upward and downward penetrations of pile, and the pile–soil sliding characteristics are considered. An analytical expression for the settlement calculation of semi-rigid pile composite foundation under embankment loading is derived by using the differential equation for pile–soil load transfer of the unit cell.

(2)

The validity of the settlement calculation formula of semi-rigid pile composite foundation is verified through in-site measurement and model tests. The theoretical calculation results are in good agreement with the results of field tests and model tests.

(3)

The test results show that the settlement stability time of the ultra-soft soil composite foundation is longer, and the settlement is large. Through model tests, it was found that the semi-rigid pile end is penetrated in the ultra-soft soil composite foundation. The relationship between pile end resistance and pile end penetration is hyperbolic.

(4)

Through the analysis of the influencing factors of the different compression moduli of the underlying layer and pile diameters on the pile end penetration, it was found that the compression modulus of the underlying layer has a great influence on the pile end penetration. The lower the compression modulus of the underlying layer, the larger the pile end penetration. The larger the pile diameter, the smaller the pile end penetration.

(5)

In the design work for composite foundations, increasing the pile diameter and placing the pile end in a high compression modulus soil layer can be used to reduce the pile end penetration, so as to reduce the compression deformation of the surrounding soil.